Theoretical Astrophysics

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Theoretical Astrophysics
Part 3

• Bram Achterberg
• a.achterberg_at_astro.uu.nl
• http//www.astro.uu.nl/achterb/astrophysics

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Kelvin-Helmholtz Instability

• An example of a fluid system
• that is not stable against small perturbations

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Numerical Simulation of the Kelvin-Helmholtz
Instabilityin an astrophysical jet
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Basic situation two streaming fluidsseparated
by a sharp boundary
z 0
After a small perturbation
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Applications in Astrophysics
Interaction of the Earths magnetosphere And the
Solar Wind
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Wiggles in parsec-scale jets of Active Galaxies
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General approach perturbation analysis of the
dynamical equations for a fluid
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A quick and dirty derivation
Equation of motion
Small perturbation
Linearized equation For the perturbed motion
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Equation of motion for small perturbations
Different equations for the two half-spaces!
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Incompressible fluctuations no sound waves!
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Incompressible fluctuations no sound waves!
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Incompressible fluctuations no sound waves!
Pressure satisfies Poissons equation!
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Remember two streaming fluidsseparated by a
sharp boundary,situation is uniform in x and y
directions!
z 0
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Solution for pressure in both fluids
Wave-like solution
Poissons equation
At fluid interface z0 pressure must be the same
on both sides!
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Solution for pressure in both fluids
Wave-like solution
At fluid interface z0 pressure must be the same
on both sides!
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Solution for pressure in both fluids
Wave-like solution
At fluid interface z0 pressure must be the same
on both sides!
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Amplitudes from Equation of Motion
Wave-like solution
Equation of motion in component form for lower
fluid
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Solution
Amplitude in lower fluid
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Solution
Amplitude in lower fluid
Upper fluid by analogy!
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Physical boundary condition at interface
displacement must match!
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Boundary condition yields dispersion relation
between ? and k !
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Boundary condition yields dispersion relation
between ? and k !
Always a growing solution the system is unstable!
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time
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Compressible case finite sound speed effects
Simplest possible situation two
counterstreaming but otherwise identical fluids
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Very symmetric KH Dispersion Relation
Incompressible case recovered for infinite sound
speed
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The compressible case effect of finite sound
speed
unstable
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Numerical simulation Of the KH Instability
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The Rayleigh-Taylor Instability
The inability of a light fluid to support a heavy
fluid against gravity
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• What will we do??
• Perturb the interface at z0

2. Look for wave-like solutions to the
equation of motion

• Find the frequency ? of the
• perturbations
• 4. Look for solutions with Im(?)gt0

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Numerical simulation Of RT Instability inside a
SN1A nuclear flame (Bell et al 2004)
g
Hot burned material less dense!
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Calculation is analogous to KH case
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Solution for pressure in both fluids
Wave-like solution
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Conditions at the interface (z0)

• Displacement the same
• on both sides of interface

2. Pressure jump due to gravity and the
density jump
From the equation of motion
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Explanation of the pressure jump
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Dirac delta-function
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Resulting dispersion relation for the frequency
of the RT Instability